5.14: Practice problems

5.14 Practice problems

5.1. Determine type of Chi-Square test Determine which type of test is appropriate in each situation – Independence or Homogeneity?

5.1.1. Concerns over diseases being transmitted between birds and humans have led to many areas developing monitoring plans for the birds that are in their regions. The duck pond on campus at MSU-Bozeman is a bit like a night club for the birds that pass through Bozeman.

1. Suppose that a researcher randomly samples 20 ducks at the duck pond on campus on 4 different occasions and records the number ducks that are healthy and number that are sick on each day. The variables in this study are the day of measurement and sick/healthy.
2. In another monitoring study, a researcher goes to a wetland area and collects a random sample from all birds present on a single day, classifies them by type of bird (ducks, swans, etc.) and then assesses whether each is sick or healthy. The variables in this study are type of bird and sick/healthy.

5.1.2. Psychologists performed an experiment on 48 male bank supervisors attending a management institute to investigate biases against women in personnel decisions. The supervisors were asked to make a decision on whether to promote a hypothetical applicant based on a personnel file. For half of them, the application file described a female candidate; for the others it described a male.

5.1.3. Researchers collected data on death penalty sentencing in Georgia. For 243 crimes, they categorized the crime by severity from 1 to 6 with Category 1 comprising barroom brawls, liquor-induced arguments, lovers’ quarrels, and similar crimes and Category 6 including the most vicious, cruel, cold-blooded, unprovoked crimes. They also recorded the perpetrator’s race. They wanted to know if there was a relationship between race and type of crime.

5.1.4. Epidemiologists want to see if Vitamin C helped people with colds. They would like to give some patients Vitamin C and some a placebo then compare the two groups. However, they are worried that the placebo might not be working. Since vitamin C has such a distinct taste, they are worried the participants will know which group they are in. To test if the placebo was working, they collected 200 subjects and randomly assigned half to take a placebo and the other half to take Vitamin C. 30 minutes later, they asked the subjects which supplement they received (hoping that the patients would not know which group they were assigned to).

5.1.5. Is zodiac sign related to GPA? 300 randomly selected students from MSU were asked their birthday and their current GPA. GPA was then categorized as < 1.50 = F, 1.51-2.50 = D, 2.51 - 3.25 = C, 3.26-3.75 = B, 3.76-4.0 = A and their birthday was used to find their zodiac sign.

5.1.6. In 1935, the statistician R. A. Fisher famously had a colleague claim that she could distinguish whether milk or tea was added to a cup first. Fisher presented her, in a random order, 4 cups that were filled with milk first and 4 cups that were filled with tea first.

5.1.7. Researchers wanted to see if people from Rural and Urban areas aged differently. They contacted 200 people from Rural areas and 200 people from Urban areas and asked the participants their age (<40, 41-50, 51-60, >60).

5.2. Data is/are analysis The FiveThirtyEight Blog often shows up with interesting data summaries that have general public appeal. Their staff includes a bunch of quants with various backgrounds. When starting their blog, they had to decide on the data is/are question that we introduced in Section 2.1. To help them think about this, they collected a nationally representative sample that contained three questions about this. Based on their survey, they concluded that

Figure 5.27: Tableplot of data from “data-is-vs-data-are” survey, sorted by “CareAbout” responses.

Relevant to the interests of FiveThirtyEight in particular, we also asked whether people preferred using “data” as a singular or plural noun. To those who prefer the plural, I’ll put this in your terms: The data are pretty conclusive that the vast majority of respondents think we should say “data is.” The singular crowd won by a 58 percentage-point margin, with 79 percent of respondents liking “data is” to 21 percent preferring “data are.” But only half of respondents had put any thought to the usage prior to our survey, so it seems that it’s not a pressing issue for most.

This came from a survey that contained questions about which is the correct usage, (isare), have you thought about this issue (thoughtabout) with levels Yes/No, and do you care about this issue (careabout) with four levels from Not at all to A lot. The following code loads their data set after missing responses were removed, does a little re-ordering of factor levels using the fct_relevel function to help make the results easier to understand, and makes a tableplot (Figure 5.27) to get a general sense of the results including information on the respondents’ gender, age, income, and education.

library(readr)
csd <- read_csv("http://www.math.montana.edu/courses/s217/documents/csd.csv")

library(tabplot)
# Need to make it explicit that these are factor variables and reorder 
# factor levels to be in "correct" order using fct_relevel:
csd <- csd %>% mutate(careabout = factor(careabout), 
                      careabout = fct_relevel(careabout,"Not at all", "Not much", 
                                              "Some", "A lot"),
                      Education = factor(Education),
                      Education = fct_relevel(Education, 
                                            levels(Education)[c(4,3,5,1,2)]),
                      Household.Income = factor(Household.Income),
                      Household.Income = fct_relevel(Household.Income, 
                                            levels(Household.Income)[c(1,4,5,6,2,3)])
                      ) 

# Sorts plot by careabout responses
tableplot(csd, select = c(isare, careabout, thoughtabout, Gender,
                 Age, Household.Income, Education), sortCol = careabout,
          pals = list("BrBG")) 

5.2.1. If we are interested in the variables isare and careabout, what sort of test should we perform?

5.2.2. Make the appropriate plot of the results for the table relating those two variables relative to your answer to 5.8.

5.2.3. Generate the contingency table and find the expected cell counts, first “by hand” and then check them using the output. Is the parametric procedure appropriate here? Why or why not?

5.2.4. Report the value of the test statistic, its distribution under the null, the parametric p-value, and write a decision and conclusion, making sure to address scope of inference.

5.2.5. Make a mosaic plot with the standardized residuals and discuss the results. Specifically, in what way do the is/are preferences move away from the null hypothesis for people that care more about this?

We might be fighting a losing battle on “data is a plural word”,

but since we are in the group that cares a lot about this, we are going to

keep trying…

Figure 5.28: Stacked bar chart of the close calls/not (overtakes less than or equal to 100 cm or not) by outfit.

5.3. Overtake close calls by outfit analysis We can revisit the car overtake passing distance data from Chapter 3 and to focus in on the “close calls”. The following code uses the ifelse function to create the close call/not response variable. It works to create a two-category variable where the first category (close) is encountered when the condition is true (Distance <= 100, so the passing distance was less than or equal to 100 cm) from the “if” part of the function (if Distance is less than or equal to 100 cm, then “close”) and the “else” is the second category (when the Distance was over 100 cm) and gets the category of notclose. The factor function is applied to the results from ifelse to make this a categorical variable for later use. Some useful code and a stacked bar chart in Figure 5.28 is provided.

dd <- read_csv("http://www.math.montana.edu/courses/s217/documents/Walker2014_mod.csv")

dd <- dd %>% mutate(Condition = factor(Condition),
                    Condition2 = reorder(Condition, Distance, FUN = mean), 
                    Close = factor(ifelse(Distance <= 100, "close", "notclose"))
                    )

plot(Close ~ Condition2, data = dd)

table1 <- tally(Close ~ Condition2, data = dd)

chisq.test(table1)

## 
##  Pearson's Chi-squared test
## 
## data:  table1
## X-squared = 30.861, df = 6, p-value = 2.695e-05

5.3.1. This is a Homogeneity test situation. Why?

5.3.2. Perform the 6+ steps of the hypothesis test using the provided results.

5.3.3. Explain how these results are consistent with the One-Way ANOVA test but also address a different research question.

References

Dayton, C. Mitchell. 1998. Latent Class Scaling Analysis. Thousand Oaks, CA: SAGE Publications.

Linzer, Drew, and Jeffrey Lewis. 2011. “poLCA: An R Package for Polytomous Variable Latent Class Analysis.” Journal of Statistical Software 42 (10): 1–29.

Linzer, Drew, and Jeffrey Lewis. 2022. poLCA: Polytomous Variable Latent Class Analysis. https://github.com/dlinzer/poLCA.

Lumley, Thomas. 2021. Survey: Analysis of Complex Survey Samples. http://r-survey.r-forge.r-project.org/survey/.

Meyer, David, Achim Zeileis, and Kurt Hornik. 2022. Vcd: Visualizing Categorical Data. https://CRAN.R-project.org/package=vcd.

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98. Install the tabplot package from the authors’ github repository using library(remotes); remotes::install_github("mtennekes/tabplot") if you haven’t already done so.↩︎
99. While randomization is typically useful in trying to “equalize” the composition of groups, a possible randomization of subjects to the groups is to put all the males into the treatment group. Sometimes we add additional constraints to randomization of subjects to treatments to guarantee that we don’t get stuck with an unusual and highly unlikely assignment like that. It is important at least to check the demographics of different treatment groups to see if anything odd occurred.↩︎
100. The vertical line, “|”, in ~ y | x is available on most keyboards on the same key as “\”. It is the mathematical symbol that means “conditional on” whatever follows.↩︎
101. Technically this is a “spineplot” as it generalizes the stacked bar chart based on the proportion of the total in each vertical bar.↩︎
102. Standardizing involves dividing by the standard deviation of a quantity so it has a standard deviation 1 regardless of its original variability and that is what is happening here even though it doesn’t look like the standardization you are used to with continuous variables.↩︎
103. Note that in smaller data sets to get results as discussed here, use the correct = F option. If you get output that contains “...with Yate's continuity correction”, a slightly modified version of this test is being used.↩︎
104. Here it allows us to relax a requirement that all the expected cell counts are larger than 5 for the parametric test (Section 5.6).↩︎
105. Doctors are faced with this exact dilemma – with little more training than you have now in statistics, they read a result like this in a paper and used to be encouraged to focus on the p-value to decide about treatment recommendations. Would you recommend the treatment here just based on the small p-value? Would having Figure 5.11 to go with the small p-value help you make a more educated decision? Recommendations for users of statistical results are starting to move past just focusing on the p-values and thinking about the practical importance and size of the differences. The potential benefits of a treatment need to be balanced with risks of complications too, but that takes us back into discussing having multiple analyses in the same study (treatment improvement, complications/not, etc.).↩︎
106. Independence tests can’t be causal by their construction. Homogeneity tests could be causal or just associational, depending on how the subjects ended up in the groups.↩︎
107. A stratified random sample involves taking a simple random sample from each group or strata of the population. It is useful to make sure that each group is represented at a chosen level (for example the sample proportion of the total size). If a simple random sample of all schools had been taken, it is possible that a level could have no schools selected.↩︎